q-theta function

In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series. It is given by

$\theta (z;q):=\prod _{n=0}^{\mathrm{\infty }}(1-q^{n}z)(1-q^{n+1}/z)$

where one takes 0 ≤ |q| < 1. It obeys the identities

$\theta (z;q)=\theta (\frac{q}{z};q)=-z\theta (\frac{1}{z};q).$

It may also be expressed as:

$\theta (z;q)=(z;q{)}_{\mathrm{\infty }}(q/z;q{)}_{\mathrm{\infty }}$

where $(\cdot \cdot {)}_{\mathrm{\infty }}$ is the q-Pochhammer symbol.

See also

• elliptic hypergeometric series
• Jacobi theta function
• Ramanujan theta function